Free subgroups in almost subnormal subgroups of general skew ...

arXiv:1602.03639v1 [math.RA] 11 Feb 2016

FREE SUBGROUPS IN ALMOST SUBNORMAL SUBGROUPS OF GENERAL SKEW LINEAR GROUPS NGUYEN KIM NGOC, MAI HOANG BIEN, AND BUI XUAN HAI

Abstract. Let D be a weakly locally finite division ring and n a positive integer. In this paper, we investigate the problem on the existence of non-cyclic free subgroups in non-central almost subnormal subgroups of the general linear group GLn (D). Further, some applications of this fact are also investigated. In particular, all infinite finitely generated almost subnormal subgroups of GLn (D) are described.

1. Introduction and preliminaries Let G be a group and H be a subgroup of G. According to Hartley [18], we say that H is almost subnormal in G, and write H asn G for short, if there is a family of subgroups H = Hr ≤ Hr−1 ≤ · · · ≤ H1 = G of G such that for each 1 < i ≤ r, either Hi is normal in Hi−1 or Hi has finite index in Hi−1 . We call such a series of subgroups an almost normal series in G. In this paper, we study the problem on the existence of non-cyclic free subgroups in almost subnormal subgroups of the general linear group GLn (D) over a division ring D and the related problems. The question on the existence of non-cyclic free subgroups in linear groups over a field was studied by Tits in [26]. The main theorems of Tits assert that in the characteristic 0, every subgroup of the general linear group GLn (F ) over a field F either contains a non-cyclic free subgroup or it is soluble-by-finite, and the same conclusion for finitely generated subgroups in the case of prime characteristic. This famous result of Tits is now often referred as Tits’ Alternative. The question of whether Tits’ Alternative would remain true for skew linear groups was posed by S. Bachsmuth at the Second International Conference on the Theory of Groups (see [2, p. 736]). In [22], Lichtman has proved that there exists a finitely generated group which is not soluble-by-finite and does not contain a non-cyclic free subgroup, but whose group ring over any field can be embedded in a division ring of quotients. Therefore, Tits’ Alternative fails even for matrices of degree one, i.e. for D∗ = GL1 (D), where D is a non-commutative division ring. In [22], Lichtman remarked that it is not known whether the multiplicative group of a non-commutative division ring contains a non-cyclic free subgroup. In [12], Gon¸calves and Mandel posed more general question: whether a non-central subnormal subgroup of the multiplicative group of a division ring contains a non-cyclic free subgroup? This question was studied by several authors. Gon¸calves [10] proved that the multiplicative group Key words and phrases. division rings; linear groups; almost subnormal subgroups; non-cyclic free subgroups; generalized group identity. 2010 Mathematics Subject Classification. 16K20, 16K40, 16R50. 1

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NGUYEN KIM NGOC, MAI HOANG BIEN, AND BUI XUAN HAI

D∗ of a division ring D with center F contains a non-cyclic free subgroup if D is centrally finite, that is, D is a finite dimensional vector space over F . The same result was obtained by Reichstein and Vonessen in [28] if F is uncountable and there exists a non-central element a in D which is algebraic over F . Later, Chiba [7] proved the same result but without the assumption on the existence of such an element a in D. In [11], Gon¸calves proved that any non-central subnormal subgroup of D∗ contains a non-cyclic free subgroup provided D is centrally finite. Recently, B. X. Hai and N. K. Ngoc [16] proved the same result for weakly locally finite division rings. Recall that a division ring D is called weakly locally finite if every finite subset of D generates a centrally finite division subring. It was proved that every locally finite division ring is weakly locally finite, and there exist infinitely many weakly locally finite division rings that are not even algebraic over their centers (see [16] and [17]), so they are not locally finite. Logically, it is natural to carry over the results above for subnormal subgroups of GL1 (D) to that of GLn (D), n ≥ 2 (see [14, 24, 30]). In the present paper, we investigate the question on the existence of free subgroups in almost subnormal subgroups of the group GLn (D) with n ≥ 1 and D is a division ring unnecessarily commutative. Note that in [19, Example 8], Hazrat and Wadsworth gave the examples of division rings whose multiplicative groups contains non-normal maximal subgroups of finite index. Hence, the existence of almost subnormal subgroups in D∗ := GL1 (D) has no doubt. Concerning the group GLn (D), n ≥ 2, we shall prove that if D is infinite then every almost subnormal subgroup of GLn (D) is normal (see Theorem 3.3 in the text). However, we shall continue to use “almost subnormal” instead of “normal” to compare the results with the corresponding ones in the case n = 1. All symbols and notation we use in this paper are standard. In particular, if A is a ring or a group, then Z(A) denotes the center of A. If D is a division ring with center F and S is a subset of D, then F (S) denotes the division subring of D generated by the set F ∪ S. We say that F (S) is the division subring of D generated by S over F . Finally, D′ := [D∗ , D∗ ] is the commutator subgroup of D∗ . The following lemma which will be used frequently in this paper, is almost evident, so we omit its proof. Lemma 1.1. Let H be an almost subnormal subgroup of a subgroup G. If N is a subgroup of G containing H, then H is an almost subnormal subgroup of N . 2. Almost subnormal subgroups with generalized group identities Let G be a group with center Z(G) = { a ∈ G | ab = ba for any b ∈ G }. An expression α2 αt 1 w(x1 , x2 , · · · , xm ) = a1 xα i1 a2 xi2 · · · at xit at+1 , where t, m are positive integers, i1 , i2 , · · · , it ∈ { 1, 2, · · · , m }, a1 , a2 , · · · , at+1 ∈ G and α1 , α2 , · · · , αt ∈ Z\{0}, is called a generalized group monomial over G if whenever ij = ij+1 with αj αj+1 < 0 (j ∈ {1, 2, · · · , t − 1}), then aj 6∈ Z(G) (see [27]). Moreover, if one has ij 6= ij+1 whenever αj αj+1 < 0, then we say that w is a strict generalized group monomial over G. If G = {1}, then we simply call w a group monomial. It is clear that a group monomial is a strict generalized group monomial. Let H be a subgroup of G. We say that H satisfies the generalized group identity w(x1 , x2 , · · · , xm ) = 1 if w(c1 , c2 , · · · , cm ) = 1 for any c1 , c2 , · · · , cm ∈ H. In

FREE SUBGROUPS IN ALMOST SUBNORMAL SUBGROUPS

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particular, we say that H satisfies a group identity (resp. strict generalized group identity) w = 1 if w is a group monomial (resp. strict generalized group monomial) and w(c1 , c2 , · · · , cm ) = 1 for any c1 , c2 , · · · , cm ∈ H. In this section, we prove some properties of almost subnormal subgroups with generalized group identities we need for the next study. We begin with the following useful lemma. Lemma 2.1. Let G be a group and assume that H is a non-central almost subnormal subgroup of G. If H satisfies a generalized group identity over G, then so does G. Proof. Assume that H is a non-central almost subnormal subgroup of G satisfying a generalized group identity over G. Let H = Hr ≤ Hr−1 ≤ · · · ≤ H1 = G be an almost normal series in G. To prove the lemma, that is, to prove that H1 satisfies a generalized group identity, it suffices to prove that Hr−1 satisfies a generalized group identity over G. Let w(x1 , x2 · · · , xm ) = a1 xni11 a2 xni22 · · · at xnitt at+1 be a generalized group identity of H over G. By replacing xi = yi yi+m , we get u(y1 , y2 , · · · , y2m ) = w(y1 y1+m , y2 y2+m , · · · , ym y2m ) = b1 yjδ11 b2 yjδ22 · · · bs yjδss bs+1 , where δi ∈ {−1, 1}, which is also a generalized group identity of H. Hence, without loss of generality, we can assume that in w, the powers ni ∈ {−1, 1} for any 1 ≤ i ≤ t. There are two cases to examine: Case 1. Hr has finite index k in Hr−1 : Then, for any c1 , c2 , · · · , cm ∈ Hr−1 , we have ck1 , ck2 , · · · , ckm ∈ Hr . By hypothesis, w(ck1 , ck2 , · · · , ckm ) = 1. But this means that Hr−1 satisfies the identity w(xk1 , xk2 , · · · , xkm ) = 1. Case 2. Hr is normal in Hr−1 : Since H = Hr is non-central, there exists a non-central element a ∈ Hr . Replacing xj by xj ax−1 for any 1 ≤ j ≤ m, we get j −1 −1 w1 (x1 , x2 , · · · , xm ) = w(x1 ax−1 1 , x2 ax2 , · · · , xm axm ),

which is a generalized group monomial over G by [3, Lemma 3.2]. Since Hr is normal in Hr−1 , ci ac−1 ∈ Hr for any ci ∈ Hr−1 . Therefore, i −1 −1 w1 (c1 , c2 , · · · , cm ) = w(c1 ac−1 1 , c2 ac2 , · · · , cm acm ) = 1,

for any c1 , c2 , · · · , cm ∈ Hr−1 , which shows that w1 = 1 is a generalized group identity of Hr−1 . Therefore, the proof of the lemma is now completed. From Theorems 1, 2 in [9], it follows that for any division ring D with infinite center F , if GLn (D) satisfies a generalized group identity, then n = 1 and D = F . Recently, it was proved that this result remains true if one replaces GLn (D) by any its subnormal subgroup [5, Theorem 1.1]. Lemma 2.1 gives us the possibility to get the following strong result. Theorem 2.2. Let D be a division ring with infinite center F and assume that G is an almost subnormal subgroup of GLn (D). If G satisfies a generalized group identity over GLn (D), then G is central.

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Proof. If G is non-central, then by Lemma 2.1, GLn (D) satisfies a generalized group identity. In view of [9], n = 1 and D is commutative, so G is central, a contradiction. Thus, G is central. Corollary 2.3. Let D be a division ring with infinite center F and assume that G is an almost subnormal subgroup of GLn (D). If G is abelian, then G is central. Proof. If G is abelian, then G satisfies the group identity xyx−1 y −1 = 1. So, by Theorem 2.2, G is central. 3. Almost subnormal subgroups of GLn (D) is normal Recall that a field K is called locally finite if every subfield generated by finitely many elements of K is finite. Hence, K is locally finite if and only if its prime subfield P is a finite field and K is algebraic over P . In [30], there is the following theorem. Theorem A. Let D be a division ring that is not a locally finite field and let n > 1 be an integer. If N is non-central normal subgroup of GLn (D) then N contains a non-cyclic free subgroup. The aim of this section is to prove that if D is an infinite division ring and n ≥ 2, then every almost subnormal subgroup of GLn (D) is normal in GLn (D). Hence, according to Theorem A, the problem on the existence of non-cyclic free subgroups in almost subnormal subgroups of general skew linear groups reduced to that in skew linear groups of degree 1. To prove this fact, we need the following some results. Theorem 3.1. Let D be a division ring, n a natural number and N a non-central subgroup of GLn (D). Suppose that either n ≥ 3 or that n = 2 but that D contains at least four elements. If xN x−1 ⊆ N for any x ∈ SLn (D), then N contains SLn (D). In particular, a non-central subgroup of GLn (D) is normal in GLn (D) if and only if it contains SLn (D). Proof. The proof follows from Theorem 4.7 and Theorem 4.9 in [1].

Lemma 3.2. Let D be a division ring and n > 1. Then, the special linear group SLn (D) satisfies a group identity if and only if D is finite. Proof. If D is finite, then SLn (D) is finite, so SLn (D) satisfies a group identity. Assume that D is infinite. Let K be a maximal subfield of D. If K is finite, then dimF D < ∞ by [21, (15.8)] which implies that D is finite, a contradiction. Therefore, K is infinite. Suppose that w(x1 , x2 , · · · , xm ) = 1 is a group identity of SLn (D). Then, GLn (D) satisfies the group identity −1 −1 w1 (y1 , y2 , · · · , y2m ) = w(y1 y2 y1−1 y2−1 , y3 y4 y3−1 y4−1 , · · · , y2m−1 y2m y2m−1 y2m ) = 1.

In view of Theorem 2.2, GLn (D) is central, a contradiction.

Let G be a group and H a subgroup of G. Denote by CoreG (H) the core of H in G, that is, \ CoreG (H) = xHx−1 . x∈G

It is well known that CoreG (H) is the largest normal subgroup of G which is contained in H. Moreover, if H is of finite index in G, then so is CoreG (H).


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The following theorem is the main result of this section. Theorem 3.3. Let D be an infinite division ring and n ≥ 2. Assume that N is a non-central subgroup of G := GLn (D). The following conditions are equivalent: (1) N is almost subnormal in G. (2) N is subnormal in G. (3) N is normal in G. (4) N contains SLn (D). Proof. The implications (4) ⇒ (3) ⇒ (2) ⇒ (1) are trivial. Now we will show that (1) implies (4). Assume that N is a non-central almost subnormal subgroup of G and N = Nr ≤ Nr−1 ≤ · · · ≤ N1 ≤ N0 = G is an almost normal series of N in G. We shall prove that Ni is normal in G for all 1 ≤ i ≤ r by induction on i. Assume that N1 has finite index in G. Then, CoreG (N1 ) is a normal subgroup, say of finite index m in G and it is contained in N1 . If CoreG (N1 ) is central, then xm y m x−m y −m = 1 for any x, y ∈ G. In view of Lemma 3.2, D is finite, a contradiction. Thus, CoreG (N1 ) is non-central normal subgroup of G. By Theorem 3.1, SLn (D) ⊆ CoreG (N1 ) ⊆ N1 . Assume that j > 1 and Nj contains SLn (D). We must prove that Nj+1 also contains SLn (D). Indeed, assume that Nj+1 has finite index in Nj . Then, the subgroup CoreNj (Nj+1 ) is normal in Nj of finite index, say k, and it is contained in Nj+1 . Assume that CoreNj (Nj+1 ) is central. Then, xk y k x−k y −k = 1 for any x, y ∈ Nj . In particular, SLn (D) satisfies the group identity xk y k x−k y −k = 1. In view of Lemma 3.2, D is finite, a contradiction. Hence, CoreNj (Nj+1 ) is noncentral normal subgroup of Nj . Therefore, xCoreNj (Nj+1 )x−1 ⊆ CoreNj (Nj+1 ) for any x ∈ SLn (D) ⊆ Nj . In view of Theorem 3.1, CoreNj (Nj+1 ) contains SLn (D) and so does Nj+1 . The implication (1) ⇒ (4) is proved, and so the proof of the theorem is now complete. Remark 3.4. Theorem 3.3 no longer holds if D is a finite field. Indeed, let D = Fq be a finite field with q elements and consider the projective special linear group PSL(n, q) which is different from the groups PSL(2, 2) and PSL(2, 3). Then, it is well-known that PSL(n, q) is a simple group. Assume that k is a prime divisor of |PSL(n, q)| and H is the inverse image of a subgroup of order k in PSL(n, q) via the natural homomorphism SL(n, q) −→ PSL(n, q). Then, H is a non-central proper subgroup of SL(n, q). By Theorem 3.1, H is not normal in GL(n, q). Hence, H is an almost subnormal subgroup of GL(n, q) which is not normal in GL(n, q). 4. Non-cyclic free subgroups in non-commutative division rings As we have mentioned in the Introduction, almost subnormal subgroups that are not subnormal exist in the multiplicative group of a division ring. The aim of this section is to show that if a non-commutative division ring D is weakly locally finite, then every non-central almost subnormal subgroup of D∗ contains a non-cyclic free subgroup. Recall that a division ring D is weakly locally finite if every finite subset

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in D generates a centrally finite division subring in D. Some basic properties and the existence of non-cyclic free subgroups in weakly locally finite division rings can be seen in [8] and [16]. The following lemma is useful for our next study. Lemma 4.1. Let D be a non-commutative weakly locally finite division ring with center F and assume that G is an almost subnormal subgroup of D∗ . If G satisfies a generalized group identity, then G is central. In particular, If G is abelian, then G is central. Proof. Assume that G is non-central and G satisfies some generalized group identity w(x1 , x2 , · · · , xm ) = 1. By Lemma 2.1, D∗ satisfies some generalized group identity w′ (x1 , . . . , xm ) = a1 xt11 a2 xt22 · · · am xtmm am+1 = 1. Let x, y ∈ D such that xy 6= yx. Consider the division subring D1 of D generated by x, y and all ai . Then, D1 is non-commutative and centrally finite. Since ai ∈ D1∗ ⊆ D∗ , w′ = 1 is a generalized group identity of D1∗ . In view of Theorem 2.2, D1 is commutative, a contradiction. Hence, G is central. Theorem 4.2. Let D be a weakly locally finite division ring with center F . Then, every non-central almost subnormal subgroup of D∗ contains a non-cyclic free subgroup. Proof. Assume that G is a non-central almost subnormal subgroup of D∗ . By Lemma 4.1, G is non-abelian. Hence, there exist a, b ∈ G such that ab 6= ba. Denote by D1 a division subring of D generated by a, b. Then, D1 is centrally finite. By Lemma 1.1, N = G ∩ D1∗ is an almost subnormal subgroup of D1∗ . We claim that N contains a non-cyclic free subgroup. Indeed, if this is not the case then, by [20, Theorem 2.21], N satisfies a group identity w(x1 , x2 , . . . , xm ) = 1. Observe that the center F1 of D1 is infinite, so by Theorem 2.2, N is central. In particular, ab = ba which is a contradiction. Thus, the claim is proved, and this implies that G contains a non-cyclic free subgroup. Now combining Theorem 4.2 and Theorem A, we summary the results on the existence of non-cyclic free subgroups in almost subnormal subgroups of the general linear group over a weakly locally finite division ring. Theorem 4.3. Let D be a weakly locally finite division ring, n a natural number and N a non-central almost subnormal subgroup of GLn (D). Then, N contains a non-cyclic free subgroup if one of the following conditions satisfies: (1) D is non-commutative; (2) n = 1; (3) n ≥ 2 and D is not a locally finite field. The following proposition extends [12, Corollary 3.4]. Proposition 4.4. Let D be a division ring with center F and G be an almost subnormal subgroup of D∗ . If G\F contains a torsion element, then G contains a non-cyclic free subgroup. Proof. Assume that a ∈ G\F such that an = 1 for some positive integer n. According to [4, Proposition 2.1], there exists a centrally finite division ring D1 such that a 6∈ F1 = Z(D1 ). Using the same argument in the proof of Theorem 4.2, we see that M = G ∩ D1∗ is an almost subnormal subgroup of D1∗ . Since a ∈ M , the subgroup M is non-central. By Theorem 4.2, M contains a non-cyclic free subgroup, so does G.


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Theorem 4.5. Let D be a weakly locally finite division ring with center F and G be an almost subnormal subgroup of D∗ . If G is soluble-by-periodic, then G is central. Proof. Assume that G is non-central. Let H be a soluble normal subgroup of G such that G/H is periodic. Then, H is almost subnormal in D∗ . Since H is soluble, H is central by Lemma 4.1. Hence, for any x ∈ G, there exists a natural number nx such that xnx ∈ H ⊆ F . But this is impossible since G contains a non-cyclic free subgroup by Theorem 4.2. Note that previously in [23, Propositon 1], Lichtman proved the same result as in Theorem 4.5 for normal subgroups in centrally finite division rings. The class of weakly locally finite division rings we consider in Theorem 4.5 is very large. Indeed, in [16], it was indicated that this class strictly contains the class of locally finite division rings. Recently, in [17], we have constructed infinitely many examples of weakly locally finite division rings that are not even algebraic over the center. Now, let D be a centrally finite division ring. The following theorem gives useful characterization of a subgroup of D∗ that contains no non-cyclic free subgroups. Theorem 4.6. Let D be a centrally finite division ring and G be a subgroup of D∗ . The following conditions are equivalent: (1) G contains no non-cyclic free subgroups. (2) G is soluble-by-finite. (3) G is abelian-by-finite. (4) G satisfies a group identity. (5) G contains a soluble subgroup of finite index. (6) G contains an abelian subgroup of finite index. (7) G satisfies a strict generalized group identity. Proof. For (1) ⇔ (2)⇔ (3)⇔ (4) see [20, Theorem 2.21]. The implications (2) ⇒ (5), (3) ⇒ (6), (6) ⇒ (5), and (4) ⇒ (7) are trivial. To prove (5) ⇒ (4), assume that G contains a soluble subgroup H of finite index [G : H] = m. Since H is soluble, H satisfies a group identity w(x1 , x2 , · · · , xn ) = 1. m m Then, w(cm 1 , c2 , · · · , cn ) = 1 for any ci ∈ G, so G satisfies a group identity m m m w(x1 , x2 , · · · , xn ) = 1. Finally, the equivalence (5) ⇔ (7) follows from [27, Theorem 1]. The proof of a theorem is now complete. In [23], Lichtman have shown that for a normal subgroup G of D∗ , if there exists a non-abelian nilpotent-by-finite subgroup in G, then G contains a non-cyclic free subgroup. Recently, Gon¸calves and Passman gave another proof and an explicit construction of non-cyclic free subgroups (see [13]). In the following theorem, we consider the case when D is algebraic over its center and generalize this result for almost subnormal subgroups. Theorem 4.7. Let D be a division ring algebraic over its center F and assume that G is an almost subnormal subgroup of D∗ . If G contains a non-abelian nilpotentby-finite subgroup, then G contains a non-cyclic free subgroup. Proof. Let N be a non-abelian nilpotent-by-finite subgroup of G. Then, there exists a nilpotent normal subgroup A of N such that [N : A] = m.

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Case 1. A is non-abelian: Since A is nilpotent, there exist x, y ∈ A such that 1 6= z = y −1 x−1 yx, zx = xz, zy = yz. Let D1 be a division subring of D generated by x, y. By [23, Lemma 1], D1 is centrally finite. Using the same argument as in the proof of Theorem 4.2, we can conclude that M = G ∩ D1∗ is a non-abelian almost subnormal subgroup in D1∗ . By Theorem 4.2, M contains a non-cyclic free subgroup. Case 2. A is abelian: Let D1 be a division subring of D generated by F and N , and F1 = Z(D1 ). Since [N : A] = m, D1 is finite dimensional over subfield F (A). By [6], D1 is centrally finite. Using the same argument as in the proof of Theorem 4.2, we conclude that M = G ∩ D1∗ is a non-abelian almost subnormal subgroup in D1∗ . By Theorem 4.2, M contains a non-cyclic free subgroup. 5. Finitely generated almost subnormal subgroups of GLn (D) In this section, we investigate finitely generated subgroups of GLn (D) with some additional conditions. Recall that if D = F is a field then Tits’ Alternative asserts that every finitely generated subgroup of GLn (F ) either contains a non-cyclic free subgroup or it is soluble-by-finite. Now, assume that G is a finitely generated subgroup of GLn (D), where D is a non-commutative division ring. It was shown in [25] that if D is centrally finite and G is a subnormal subgroup in GLn (D), then G is central. In the case when n = 1, it was proved in [15, Theorem 2.5] that if D is of type 2, then there are no finitely generated subgroups of D∗ containing the center F ∗ . Recall that a division ring D is said to be of type 2 if the division subring F (x, y) of D generated over its center F by any two elements x, y ∈ D is a finite-dimensional vector space over F . The aim of this section is to carry over these results for almost subnormal subgroups of GLn (D), where D is a weakly locally finite division ring. Recall that Theorem 3.3 implies that any almost subnormal subgroup of GLn (D) is normal if n ≥ 2, but we shall continue to use “almost subnormal” instead of “normal” to compare the results with the corresponding ones in the case n = 1. The following result, which is an easy consequence of Theorem 2.2 gives the characterization of finite almost subnormal subgroups in division rings. Lemma 5.1. Let D be a division ring with center F and assume that G is an almost subnormal subgroup of D∗ . If G is finite, then G is central. Proof. Let D1 = F (G) be the division subring of D generated by G over F . By Lemma 1.1, G is almost subnormal in D1∗ . If F is finite, then D1 is a field. In particular, G is abelian, so by Theorem 4.5, G is central. If F is infinite, then so is the center of D1 . Hence, in view of Theorem 2.2, G is central.

Remark 5.2. Let H = ha1 , a2 , · · · , am i be a finitely generated subgroup of GLn (D), where D is a division ring. Denote by S the set of all entries of all matrices ai , a−1 i , and by R the subring of D generated by S. Then, H is contained in GLn (R). In particular, H is contained in GLn (D1 ), where D1 is the division subring of D generated by S. This fact will be used frequently in this section.


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Let G be a subgroup of GLn (F ), where F is a field. Suppose that a ∈ GLn (F ). It is easy to see that a + xIn is non-invertible for finitely many elements x ∈ F : if a + xIn is non-invertible, then the determinate |a + xIn | of a + xIn , a polynomial of degree ≤ n in x, is 0. By the Vandermonde argument [29, Propositions 2.3.26 and 2.3.27], there are finitely many elements x ∈ F such that |a + xIn | = 0. Now assume that H is an almost subnormal subgroup of G, and H = Hr ≤ Hr−1 ≤ · · · ≤ H1 = G is an almost normal series of subgroups of G. For any a, b ∈ H and x ∈ F such that b + xIn is invertible, put c1 (a, b, x) := (b + xIn )a(b + xIn )−1 , and for 1 < i ≤ r, we define ci inductively as the following: if Hi is normal in Hi−1 , then ci (a, b, x) := ℓi ci−1 bc−1 i−1 , otherwise ci (a, b, x) := ci−1 , where ℓi is the index of Hi in Hi−1 . Lemma 5.3. Let ci (a, b, x) be as above. Then, ci = (b + xIn )wi (a, b)(b + xIn )−1 , where wi (a, b) is a reduced word in a, b, a−1 , b−1 which begins and ends by a or a−1 . Proof. We prove the lemma by induction in 1 ≤ i ≤ r. If i = 1, then w1 (a, b) = (b + xIn )−1 c1 (b + xIn ) = a. Assume that ci = (b + xIn )wi (a, b)(b + xIn )−1 , where a reduced word wi (a, b) begins and ends by a or a−1 for any i ≤ 1. We have to show that ci+1 has a same property, that is, ci+1 (a, b, x) = (b + xIn )wi+1 (a, b)(b + xIn )−1 with a reduced word wi+1 (a, b) ending by a or a−1 . Indeed, there are two cases to examine. Case 1. If Hi+1 is normal in Hi , then ci+1 (a, b, x) = ci bc−1 i = ((b + xIn )wi (a, b)(b + xIn )−1 )b((b + xIn )wi (a, b)(b + xIn )−1 )−1 = (b + xIn )wi (a, b)bwi (a, b)−1 (b + xIn )−1 = (b + xIn )wi+1 (a, b)(b + xIn )−1 , where wi+1 = wi (a, b)bwi (a, b)−1 . Case 2. If Hi+1 has finite index ri+1 in Hi , then ci+1 (a, b, x) = ci (a, b, x)ℓi+1 = ((b + xIn )wi (a, b)(b + xIn )−1 )ℓi+1 = (b + xIn )wi (a, b)ℓi+1 (b + xIn )−1 = (b + xIn )wi+1 (a, b)(b + xIn )−1 , where wi+1 = wi (a, b)ℓi+1 . The proof of the lemma is now complete.

Notice that in the proof of main results in [25], the authors considered two cases n = 1 and n > 1 separately with two difference arguments. By modifying the proof of the case n = 1 in [25], we will prove our main result as the following for the arbitrary case. Theorem 5.4. Let D be a weakly locally finite division ring, and assume that G is an infinite almost subnormal subgroup of GLn (D). If G is finitely generated, then G is central. Proof. Assume by contrary that G is non-central finitely generated subgroup of GLn (D). Since D is weakly locally finite, by Remark 5.2, without loss of generality, we may assume that D is centrally finite with [D : F ] = t < ∞, where F is the center of D.

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We first claim that G contains a non-cyclic free subgroup by showing that D and G satisfy the requirements of Theorem 4.3. Indeed, if char(F ) = p > 0 and D = F is a field algebraic over its prime subfield, then every element of GLn (F ) is torsion. By Schur’s Theorem [21, Theorem (9.9), p. 154], G is finite that contradicts to the hypothesis. Therefore, the claim is proved. Clearly the group GLn (D) may be viewed as a subgroup of GLnt (F ), so G is also a subgroup of GLnt (F ). By Remark 5.2, G is a subgroup of GLnt (P (S)), where P is the prime subfield of F and S is a finite set of F . Since G is infinite, so is P (S). Let us consider two cases. Case 1. char(P ) > 0: Let S = {α1 , α2 , . . . , αh }. If all elements of S are algebraic over P , then P (S) is finite, a contradiction. Let i0 be the largest index such that α := αi0 is not algebraic over P . If K = P (α1 , α2 , . . . , αi0 −1 ) and L = K(α), then [P (S) : L] = s < ∞. Therefore, G may be viewed as a subgroup of GLnts (L). Recall that α is not algebraic over K, so L can be considered as the field of fractions of K[α]. Again by Remark 5.2, we can find a finite subset um (α) u1 (α) u2 (α) , ,··· , T = v1 (α) v2 (α) vm (α) of L such that GLnts (K[α][T ]) contains G. Note that v1 (α), u1 (α), · · · , vm (α), um (α) are elements of K[α] such that vi (α) and ui (α) are co-prime for any i. Let a, b ∈ G be two elements such that ha, bi is a non-abelian free subgroup of G and let G = Gr ⊆ Gr−1 ⊆ · · · ⊆ G1 = GLn (D) be an almost normal series of G in GLnts (K[α][T ]). Observe that b + xInts is non-invertible for finitely many elements x ∈ K[α][T ]. Now, take an x ∈ K[α][T ] such that b + xInt is invertible. By Lemma 5.3, cr (a, b, x) = (b + xInts )wr (a, b)(b + xInts )−1 ∈ Gr = G. We claim that all the entries of cr (a, b, x) do not depend on x. Indeed, assume that there exists an entry (i, j)-th of cr (a, b, x) depending on x. Without loss of generality, we may assume that (i, j) = (1, 1). Notice that the determinant of b + xInts is a polynomial f (x) in x of degree q = nts, so the (1, 1)-th entry has the form g(x) bq xq + bq−1 xq−1 + · · · + b0 ∈ K[α][T ]. = q f (x) x + cq−1 xq−1 + · · · + c0 Assume that bq =

um+1 (a) vm+1 (a) .

Then,

g(x) f (x)

m+1 (a) − bq ∈ K[α][T ∪ { uvm+1 (a) }], and clearly, the

degree of the numerator of fg(x) (x) − bq is less than q. Hence, without loss of generality, we may assume that bq = 0, that is, bq−1 xq−1 + · · · + b0 g(x) ∈ K[α][T ]. = q f (x) x + cq−1 xq−1 + · · · + c0 Observe that c0 is the determinant of b, which is invertible element in GLnts (K[α][T ]), so c0 6= 0. Put g1 (x) c−1 0 g(x) = −1 , f1 (x) c0 f (x) −1 q −1 q−1 q−1 that is, g1 (x) = c−1 +· · ·+c−1 +· · ·+1. 0 bq−1 x 0 b0 and f1 (x) = c0 x +c0 cq−1 x Let w1 (α), · · · , wℓ (α) be all prime factors of v1 (α), u1 (α), · · · , vm (α), um (α) and put

x(α) = (w1 (α)w2 (α) · · · wl (α))p .


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Then, since the degree g1 (x) is less than f1 (x)’s when p is large enough, the degree of the denominator f1 (x(α)) in α is greater than the numerator’s which implies that there exists i such that f1 (x(α)) is a multiple of wi (α). Hence, 1 is also a multiple of wi (α) which is a contradiction. Thus, the claim is proved. Therefore, cr (a, b, x) does not depend on x. Now, one has cr (a, b, 0) = cr (a, b, y) for some y ∈ K[α]\{0} such that b + yIq is invertible. Hence, bwr (a, b)b−1 = (b + yIq )wr (a, b)(b + yIq ), equivalently, bwr (a, b)b−1 = wr′ (a, b), which is a contradiction to the fact that a, b are generators of a non-cyclic free group. Case 2. char(F ) = 0: If P (S) is not algebraic over Q, then by the same procedure as in the first part of Case 1, we conclude that the field P (S) contains a subfield L1 = K1 (β) such that [P (S) : L1 ] = s1 < ∞, where K1 is a subfield of P (S) and β is not algebraic over K1 . Now, again by the same procedure as in Case 1 with replacing K(α) by K1 (β), one has a contradiction. Therefore, the proof of the theorem is now complete.

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